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Questions tagged [complex-dynamics]

This tag is for questions relating to complex dynamics, study of dynamical systems defined by iteration of functions on complex number spaces. It was an area of research established by Fatou and Julia towards the beginning of the last century.

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Solving $f(x) = f(\frac{a + b x}{c + d x}) = f(\frac{a' + b' x}{c' + d' x})$?

How to solve the equation $$f(x) = f(\frac{a x + b}{c x + d}) = f(\frac{a'x + b'}{c'x + d'})$$ For given real $a,a',b,b',c,c',d,d'$ ? Maybe this system of equations is a bit overdetermined in its ...
mick's user avatar
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Where to find out more about escape time fractal made by $z_{n+1}=a^{z_n}$

I was messing around making a very very simple mandelbrot renderer in python, and decided to replace the z = z**2 + c in my code with ...
Kass69's user avatar
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Is a boundary of a completely invariant set is still completely invariant under an analytic function?

$R(z)$ is a rational function from the extended plane to the extended plane. Consider $D$ is a union of components of the Fatou set of $R(z)$, If $D$ is completely invariant, then how to prove that ...
zhanghaoyu_'s user avatar
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Image of a sector under a polynomial $P(z)$

Let $P(z)=a_0+a_1z+\cdots+a_dz^d$ be a polynomial where $a_i\in\mathbb{C}\setminus{\mathbb{R}}^-$ for $i=0,1,...,d$ with $a_d\neq 0$. Consider the sector $A=\{c\in\mathbb{C}:\beta\leq Arg(z)\leq \frac{...
Factorial_zero's user avatar
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56 views

How can I get this estimation?

This question is from the book Complex Dynamics by Gamelin, the theorem 2.1 's proof. Suppose $0$ is an attracting fixed point of $f$, with multiplier $\lambda$ satisfying $0<|\lambda|<1$ . How ...
zhanghaoyu_'s user avatar
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Julia set invariant under iteration of $f$

According to Falconer's Book (Fractal Geometry: Mathematical Foundations and Applications), Proposition 14.3, we have the following result: The Julia set $J=J(f)$ of $f$ is forward and backward ...
Mikeys00's user avatar
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First Order linear PDE with complex variable coefficients.

Consider the following first order linear equations: $$\partial_t u(t,x) = a(t,x)\nabla u(t,x)+b(t,x)u(t,x),u(0,x)=u_0(x).$$ If $a,b$ are real functions, this can be solved by characteristic method. ...
xinggu's user avatar
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Entire function of finite order having a deficient value

Motivation: We know that deficient values of an entire function $f$ are important because there is a connection with the singular values of $f$ and hence a connection with the Fatou set of $f$. There ...
Factorial_zero's user avatar
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1 answer
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How do I compute the action of an automorphism on the Néron-Severi group of a projective variety?

I am trying to read Dynamics of Automorphisms of Compact Complex Surfaces by Serge Cantat, and I am confused by his example of surfaces of degree (2, 2, 2) in section 2.4.6. The setup is the following:...
Zac's user avatar
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deriving/explaining the Distance Estimation Method (by gradient) for rendering Julia & Mandelbrot sets

This question is about the derivation / explanation of the Distance Estimation Method (DEM) for rendering Julia and Mandelbrot fractals. I have not succeeded in finding an explanation online so I ...
Penelope's user avatar
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Why does this distance estimator method render The Mandelbrot set incorrectly (non-divergent regions as divergent)?

I am using the following algorithm to render the Mandelbrot set and the exterior: • for each test point, calculate $c$ • initialise $z_{0}=(0+0i)$ • also initialise the gradient $dz_{0}=(0+0i)$ • ...
Penelope's user avatar
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Multiply connected Fatou component of an entire function.

This question may be trivial but still I want to know the answer. Question: Is there any necessary condition (except boundedness) for the existence of a multiply connected Fatou component of a ...
Factorial_zero's user avatar
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Why more iterations benefit deeper Mandelbrot zooms over shallow zooms?

When rendering the Mandelbrot set fractal, we set a maximum number of iterations to test each point. If the escape criteria are met within the maximum iterations, we can stop further iterations ...
Penelope's user avatar
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polynomial-like map is a ramified cover map

Recently, I am studying polynomial-like map in higher dimension, according to the book "Holomorphic Dynamical Systems" (Lecture Notes in Mathematics), page 235. Suppose $U, V$ are open ...
MATHQI's user avatar
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Example of a buried Julia component of a transcendental meromorphic function.

We know examples of buried Julia components (Definition: A Julia component is called buried if it is not contained in the boundary of any Fatou component) for rational functions. In 1998, McMullen ...
Factorial_zero's user avatar
3 votes
1 answer
140 views

Popular escape condition for calculating Mandelbrot Set, $|z|>2$, is incomplete?

The Mandelbrot set is often rendered by calculating whether a point, a complex number $c$, diverges under the iterated function $z_{n+1}=z^2_n+c$, where $z_0=0$ and $c$ is the point being tested. ...
Penelope's user avatar
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2 votes
1 answer
81 views

Equivalent description of Julia set

Let $f$ be a rational map acting on the Riemann sphere $\widehat{\mathbb{C}}$. The Julia set $\mathcal{J}_f$ is the complement of the Fatou set $\mathcal{F}_f$, defined to be the union of all open ...
user0134's user avatar
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Problem about finite grand orbits

I am attempting an exercise in Milnor's Dynamics in One Complex Variable which I have slightly rephrased below: Let $f \in \mathbb{C}(z)$ be a rational function of degree $d \geq 2$. Prove: $0, \infty$...
user0134's user avatar
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2 answers
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Complex Dynamics book Recommendation

I know the basics of Complex Analysis and Topology and I would like to learn Complex Dynamics. One book I found was Beardon's Iteration of Rational Functions. I'm not sure whether its a good book for ...
Ajin Shaji Jose's user avatar
2 votes
1 answer
162 views

Mandelbrot set Proof for Bounding Circle of Period 2 Bulb

I've been searching and I can't find a proof for the bounding circle of the Period 2 Bulb in a Mandelbrot set. Its referenced quite a bit that it is a circle with radius of $\frac{1}{4}$ and a centre ...
Joshua Finlayson's user avatar
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Finite critical points of a polynomial are preperiodic implies the Fatou set is connected and simply connected

I'm currently going through Alan Beardon's book "Iteration of Rational Functions" and I'm a little stuck on his explanation of Corollary 9.5.3. which states that "If every finite ...
OllyT777's user avatar
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Conjugating Branch Points $f=ghg^{-1}$

Consider the function $f:=ghg^{-1}$ on $\widetilde{\mathbb{C}}$ where $g$ is a homeomorphism and $h$ is a rational map. Why is it true that branch points of $h$ are transformed by $g$ to removeable ...
OllyT777's user avatar
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1 answer
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Rational Mappings of the Annulus

Suppose $R:\widetilde{\mathbb{C}} \rightarrow \widetilde{\mathbb{C}}$ where $R(A) = B$ is a rational mapping from one annulus to another. Assume that one of the components of the complement of $A$ has ...
OllyT777's user avatar
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0 answers
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Extending a covering map off an annulus

Let $R$ be a map from the Riemann sphere to itself, upon which its restriction to an annulus $A$ is a covering map to another annulus $B$. Suppose there are critical points in one of the complementary ...
OllyT777's user avatar
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Finite Number of Periodic Points in the Julia Set

I'm working through Sullivan's proof to his no wandering theorem, and in one of his sections he claims that the set of points of lowest period in the Julia set is finite. I am struggling to see why ...
OllyT777's user avatar
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1 answer
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Herman rings for polynomials

I am reading this link on complex dynamics and in Problem 12-1 it asks the reader to prove, using the Maximum Modulus Principle, that Herman rings cannot occur for polynomials. I have seen this ...
Uri Toti's user avatar
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1 answer
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Critical points of rational mappings of annuli Fatou Component

Given a rational function $R$ from the Riemann sphere to itself and an annulus fatou component $A_0$ we can create the chain $$A_0 \xrightarrow{R} A_1 \xrightarrow{R} A_2 \xrightarrow{R} ...$$ One can ...
OllyT777's user avatar
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How to prove that this polynomial inequality region is connected?

I understand that proof that the Mandelbrot Set is connected is not easy to follow and requires mathematical tools an Engineer probably doesn't have. But I got to wondering if there was a more ...
Jerry Guern's user avatar
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3 votes
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Non-zero critical value of a complex polynomial

Let $f$ be a complex polynomial of degree $d \geq 2$ with a simple root at $0$. Assume further that its other roots $z_{1},\cdots,z_{d-1}$ satisfy $\min_{1\leq i\leq d-1}(|z_{1}|,\cdots,|z_{d-1}|)=1$, ...
abeaumont's user avatar
1 vote
1 answer
52 views

Reference/translation request: Doaudy and Hubbard

Does anyone know where to find an english translation of Adrien Douady, John Hubbard - Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes) It is clearly ...
jephwack's user avatar
1 vote
0 answers
95 views

Iterating $\tanh(A x)$ and $\lim_{x \to +\infty} \tanh^{[r]}(A x) = C_r$?

Everybody who ever studied special relativity or hyperbolic trig knows this function $$\tanh(Ax)$$ for real $0 < A < 1$ $\tanh(A x)$ has a nice attracting fixpoint at $0$ and it is asymptotic to ...
mick's user avatar
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1 answer
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Inequality involving the distance to the nearest integer

i've been searching for several days trying to prove this inequality below. Let $\alpha$ an irrationnal number. First let's write for $n,j>0$ some integers, $\sigma_j^{+}(n)=1$ if $0<\{n\alpha\}&...
OdeurAtroce's user avatar
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Maclaurin series for $ f(f(z)) = \exp(z) - \exp(-z) + \exp(-z/2) - \exp(-z/5)+ \exp(-z/6) - \exp(-z/9)$

I wonder about the Maclaurin series of the analytic $f(z)$ such that $$ f(f(z)) = g(z) = \exp(z) - \exp(-z) + \exp(-z/2) - \exp(-z/5)+ \exp(-z/6) - \exp(-z/9) $$ Since zero is a fixpoint and $g'(0) = \...
mick's user avatar
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Dynamics of the map $z \mapsto z^2$

I am studying the dynamics of the map $f(z)=z^2$ on $\overline{\mathbb{C}}$, where $\overline{\mathbb{C}}$ is Riemann sphere, from the book of Alan Beardon "Iteration of rational maps" . I ...
Nirmal Rawat's user avatar
6 votes
1 answer
114 views

For a complex number on the unit circle with irrational argument z, does $z^{N!}$ ever converge?

Let $z = e^{2\pi i\alpha}$ with $\alpha \in [0, 1[$ irrational. I'm convinced the sequence $(z^{N!})_{N \in \mathbb{N}}$ has no reason to ever stabilize but I'm not sure how to prove or disprove this. ...
Pedro Lourenço's user avatar
6 votes
1 answer
198 views

Is the interior of the mandelbrot set connected?

I know that the Mandelbrot set is connected, but what about its interior? It doesn't seem intuitively like it should be, but I can't find any information online confirming this. I can think of an ...
Ali's user avatar
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1 vote
1 answer
183 views

Oscillations in Newton's fractal

I'm working on a program that draws Newton's fractal for a given polynomial. Newton's fractal is a fractal derived from Newton's root-finding method, which given some initial guess $x_0$ and function $...
zenzicubic's user avatar
1 vote
1 answer
384 views

$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?

While talking about tetration with my friend the following idea (re)occured. Equation A $$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $$ or variations of it like the weaker Equation B $$f(f(f(f(z)))) = z , ...
mick's user avatar
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0 votes
1 answer
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Persistence of periodic points

I'm currently studying some of the more deeper results in complex dynamics theory and I've come across a statement that is sometimes used in proofs seemingly without justification. Let $X$ be a ...
Maths Matador's user avatar
4 votes
1 answer
144 views

Limits of recursions like $f(n+2)=\frac{1}{f(n)} - \frac{1}{f(n+1)}$ !?

Consider the sequences $$f(0)=1,f(1)=2$$ $$f(n+2)=\frac{1}{f(n)} - \frac{1}{f(n+1)}$$ $$g(0)=1,g(1)=2,g(2)=3,g(3)=4$$ $$g(n+4)=\frac{3}{g(n)} - \frac{3}{g(n+1)} + \frac{3}{g(n+2)} - \frac{3}{g(n+3)}$$ ...
mick's user avatar
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5 votes
1 answer
112 views

How to prove this theorem for the number of components of a filled julia sets?

If one finite critical point of $f(z)$ escapes to infinity by iterating, then the filled-in Julia set of $f(z)$ consists of infinitely many components. How to prove this ? I must admit I heard this in ...
mick's user avatar
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1 vote
1 answer
160 views

Number of connected components for the filled julia set of $z^2 + c z^5$

For any polynomial map $f$ we can define the filled Julia $K$ to be closure of the complement of $ \Omega$ in $\mathbb{C}$ of the basin of infinity $$\Omega = \{z \in \mathbb{C}; f^{\circ n}(z)\...
mick's user avatar
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1 answer
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Sign of $n$ th derivative of $f(x)$?

Let $f(z)$ satisfy $f(f(z)) = \operatorname{arcsinh}(z/2)$ More precisely, we construct such an $f(z)$ by using the fixpoint at $0$ and the related Koenigs function. see : https://en.wikipedia.org/...
mick's user avatar
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1 vote
2 answers
156 views

Introducing undergraduate students to dynamical systems

In my department a course on dynamical systems is offered this semester. It is a course offered to third (out of four) year undergraduate students and it involves basic dynamics of real maps, ...
Prelude's user avatar
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1 answer
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Equicontinuous Family at a Point

I am reading the definition of equicontinuous family at a point from book called "Iteration of Rational Functions" by Alan F. Beardon. There it is written that equicontinuity of family at a ...
Nirmal Rawat's user avatar
2 votes
3 answers
137 views

Construction of Mandelbrot Set

I am doing Master's project in Complex Dynamics. Here I want to talk particularly about the Mandelbrot set. I have studied its formation and dynamics as parameter $c$ changes (which is very hand ...
Nirmal Rawat's user avatar
0 votes
1 answer
58 views

Hyperbolic Set of Extended Complex Plane

I am studying one of the research article "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets" by Mitsuhiro Shishikura. There is one section where he gave the ...
Nirmal Rawat's user avatar
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0 answers
36 views

Math function for plotting a surface that looks like crumpled paper?

Is there a way to draw a two-dimensional or three-dimensional surface that resembles this? All I know so far is that the dynamics of paper are more complicated than they sound, does this mean I could ...
blindeyes's user avatar
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1 vote
1 answer
66 views

Are there any points on the parameter plane that do not belong to any wake?

p/q-wake is the region of parameter plane enclosed by two external rays landing on the same root point on the boundary of Mandelbrot set main cardioid (period 1 hyperbolic component). Are there any ...
Adam's user avatar
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1 vote
1 answer
106 views

How to prove that the Douady-Hubbard conformal map from the exterior of Mandelbrot Set to exterior of unit disc is actually holomorphic?

I was reading the Orsay Notes on Exploring the Mandelbrot Set. (https://pi.math.cornell.edu/~hubbard/OrsayEnglish.pdf) On Page 64, it is proven that the Mandelbrot Set is connected. I understood the ...
MawnLower's user avatar

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